Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4451 |
| automorphism | Aut(G) = C22×D3×D42 (order 1536) · Inn(G) = (C2×D3)⋊C4 · Out(G) = C25 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #332 |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(5,8)(6,11)(9,10)(12,13,14,18,16,19,15,17);(2,3)(4,5,7,10)(6,8,11,9)(13,17)(14,15)(18,19)Polycyclic presentation:
2^6,3|a^2=e,b^2=c,c^2=ef,d^2=f,b^a=bd,c^a=cef,d^a=df,g^a=g^2,d^b=de39 conjugacy classes — view the full character table.
| maximal subgroup of | #68185, #68193, #68200, … |
| maximal normal subgroup of | #68185, #68193, #68200, … |
| maximal quotient of | #67477, #67563, #67564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 45);